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jackmell
#5
Dec23-11, 04:50 AM
P: 1,666
That's ok Mathwonk. I figured you are busy. Thanks for the Walker reference and the other references above. They're helpful. Also, I'm afraid I used an abuse of notation above. That's really three factors:

[tex]f(z,w)=\left(w-g(z)\right)\left(w-h(\epsilon_1z^{1/2})\right)\left(w-h(\epsilon_2 z^{1/2})\right)[/tex]

however the two factors of [itex]h(z^{1/2})[/itex] come from the same 2-valued manifold, just different single-valued determinations of it. Just thought my condenced notation was cleaner-looking.

And the expansion comes from Newton-Puiseux's Theorem:

The quotient field [itex]\mathbb{C}((z^*))[/itex] of convergent fractional power series (which I assume includes all analytic functions) is algebraically closed so for [itex]f(z,w)=a_n(z)w^n+\cdots+a_0(z)[/itex], with [itex]a_i(z)\in \mathbb{C}((z^*))[/itex], we have

[tex]f(z,w)=\prod_{i=1}^n \left(w-g_i(z^{1/k_i})\right)[/tex]

with [itex]g_i(z^{1/k_i})\in\mathbb{C}((z^*))[/itex]. One reason I knew the expansion was in terms of [itex]g(z)[/itex] and [itex]h(z^{1/2})[/itex], is because I drew it:

(Not entirely sure I'm quoting that theorem precisely. This is all very new to me.)
Attached Thumbnails
myalgebraic curve.jpg